[Paper Review] Applications of Foelner's condition to quantum groups

David Kyed, Andreas Thom|arXiv (Cornell University)|Dec 1, 2009

Advanced Operator Algebra Research19 references3 citations

TL;DR

This paper applies Følner's condition to coamenable quantum groups to establish a convergence result for Murray-von Neumann dimensions in the von Neumann algebra setting, while also deriving structural properties of the associated Hopf algebras, particularly regarding their Ore ring structure. The key contribution is the approximation of the dimension of the kernel of a certain operator via finite-dimensional representations, leveraging Følner sequences in the fusion algebra of irreducible corepresentations.

ABSTRACT

Using the Foelner condition for coamenable quantum groups we derive information about the ring theoretical structure of the Hopf algebras arising from such quantum groups, as well as an approximation result concerning the Murray von Neumann dimension associated with the corresponding the von Neumann algebra.

Motivation & Objective

• To investigate the ring-theoretic structure of Hopf algebras arising from coamenable quantum groups using Følner's condition.
• To establish an approximation result for the Murray-von Neumann dimension in the von Neumann algebra associated with a compact quantum group.
• To connect the Følner condition in the fusion algebra of irreducible corepresentations to the dimension of kernels of certain operators on $L^2({\mathbb{G}})$.
• To demonstrate that the dimension of the kernel of a representation operator on $L^2({\mathbb{G}})$ can be approximated by finite-dimensional counterparts via Følner sequences.
• To show that the dimension of the kernel of $R_T^{(2)}$ on $L^∞({\mathbb{G}})$ converges to that on $L^∞({\mathbb{G}}_i)$ as $i$ increases, under suitable conditions.

Proposed method

• The paper uses the Følner condition in the fusion algebra of irreducible corepresentations of a coamenable compact quantum group $\mathbb{G}$, defined via a finite set $F$ with small boundary relative to its size.
• It constructs a finite-dimensional approximation of the operator $R_T^{(2)}$ on $L^2({\mathbb{G}})$ by pulling back via a homomorphism $\pi_i$ from a direct limit of quantum groups $\mathbb{G}_i$.
• The key technical tool is a commutative diagram involving unitary maps $\pi_i^{(2)}$ that preserve the structure of the operator $R_T^{(2)}$ and its kernel across approximating quantum groups.
• The proof relies on estimating the difference in Murray-von Neumann dimensions using the boundary size $|\partial_S F|$ and the Haar weight $\hat{h}_{{\mathbb{G}}_i}$ on the boundary and support of $F_i$.
• It applies the inequality $|\dim_{L^\infty(\mathbb{G})}\ker(R_T^{(2)}) - \dim_{L^\infty(\mathbb{G}_i)}\ker(R_{T_i}^{(2)})| \leq 2n \frac{|\partial_S F|}{|F|}$ to show convergence.
• The convergence is established by choosing $F$ such that $|\partial_S F|/|F| < \varepsilon/(2n)$, ensuring the dimension difference is less than $\varepsilon$ for sufficiently large $i$.

Experimental results

Research questions

• RQ1 How does Følner's condition in the fusion algebra of irreducible corepresentations relate to the dimension of kernels of operators on $L^2({\mathbb{G}})$?
• RQ2 Can the Murray-von Neumann dimension of the kernel of $R_T^{(2)}$ on $L^\infty({\mathbb{G}})$ be approximated by finite-dimensional representations of quantum groups?
• RQ3 What structural properties of the Hopf algebra $\operatorname{Pol}({\mathbb{G}})$ arise from coamenability and the Følner condition?
• RQ4 Under what conditions does the kernel dimension of $R_T^{(2)}$ on $L^\infty({\mathbb{G}})$ converge to that on $L^\infty({\mathbb{G}}_i)$ as $i$ increases?
• RQ5 How does the injectivity of the map $\pi_i$ on the support and fusion products of $F$ and $S$ ensure the preservation of boundary and kernel structure?

Key findings

• The Murray-von Neumann dimension of the kernel of $R_T^{(2)}$ on $L^\infty({\mathbb{G}})$ can be approximated arbitrarily closely by the corresponding dimension on $L^\infty({\mathbb{G}}_i)$ for sufficiently large $i$, provided $\mathbb{G}_i$ is a direct limit of quantum groups with $\pi_i$ locally injective on a relevant set.
• The difference in kernel dimensions satisfies the bound $|\dim_{L^\infty(\mathbb{G})}\ker(R_T^{(2)}) - \dim_{L^\infty(\mathbb{G}_i)}\ker(R_{T_i}^{(2)})| \leq 2n \frac{|\partial_S F|}{|F|}$, which can be made arbitrarily small via the Følner condition.
• For any $\varepsilon > 0$, there exists a finite set $F \subseteq \operatorname{Irred}(\mathbb{G})$ such that $|\partial_S F|/|F| < \varepsilon/(2n)$, ensuring the convergence of the dimension approximation.
• The kernel dimension on the finite-dimensional approximation $F_i$ satisfies $\dim_F(\ker(R_T^F)) = \dim_{F_i}(\ker(R_{T_i}^{F_i}))$, preserving the structure under $\pi_i$.
• The Haar weight $\hat{h}_{{\mathbb{G}}_i}$ on the boundary $\partial_{S_i}F_i$ matches that of $\hat{h}_{{\mathbb{G}}}$ on $\partial_S F$, ensuring consistency in the dimension estimates.
• The convergence of the kernel dimension is established via a commutative diagram involving unitary maps $\pi_i^{(2)}$, which preserve the operator structure and kernel dimensions across the approximating quantum groups.